Estimating Smoothness and Optimal Bandwidth for Probability Density Functions

The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One the was introduced by Rosenblatt, another one in this paper. For case known parameter, rates mean square convergence optimal (on bandwidth) found. unknown estimation procedure parameter developed and almost surely convergency proved. sure sense these obtained. Adaptive densities given on basis constructed presented. It shown examples how parameters adaptive procedures can be chosen. Non-asymptotic asymptotic Specifically, upper bounds error a fixed sample size found their strong consistency established. Simulation results illustrate realization behavior when grows large.